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An interior-point $l_{\frac{1}{2}}$-penalty method for inequality constrained nonlinear optimization
1. | Business School, Hunan University, Changsha 410082, Hunan Province, China |
2. | Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong |
3. | School of Economics and Management, Southwest Jiaotong University, Chengdu 610031, China |
References:
[1] |
H. Y. Benson, A. Sen and D. F. Shanno, Interior-point methods for nonconvex nonlinear programming: Convergence analysis and computational performance,, , (2009). Google Scholar |
[2] |
H. Y. Benson, D. F. Shanno and R. J. Vanderbei, Interior-point methods for nonconvex nonlinear programming: Jamming and numerical testing,, Mathematical Programming, 99 (2004), 35.
doi: 10.1007/s10107-003-0418-2. |
[3] |
R. H. Byrd, G. Lopez-Calva and J. Nocedal, A line search exact penalty method using steering rules,, Mathematical Programming, 133 (2012), 39.
doi: 10.1007/s10107-010-0408-0. |
[4] |
R. H. Byrd, J. Nocedal and R. A. Waltz, Steering exact penalty methods for nonlinear programming,, Optimization Methods and Software, 23 (2008), 197.
doi: 10.1080/10556780701394169. |
[5] |
R. Chartrand and V. Staneva, Restricted isometry properties and nonconvex compressive sensing,, Inverse Problems, 24 (2008).
doi: 10.1088/0266-5611/24/3/035020. |
[6] |
L. Chen and D. Goldfarb, Interior-point $l_2$-penalty methods for nonlinear programming with strong global convergence properties,, Mathematical Programming, 108 (2006), 1.
doi: 10.1007/s10107-005-0701-5. |
[7] |
L. Chen and D. Goldfarb, On the Fast Local Convergence of Interior-point $l_2$-penalty Methods for Nonlinear Programming,, Technical report, (2006). Google Scholar |
[8] |
X. J. Chen, Smoothing methods for nonsmooth, nonconvex minimization,, Mathematical Programming, 134 (2012), 71.
doi: 10.1007/s10107-012-0569-0. |
[9] |
A. R. Conn, N. I. M. Gould, D. Orban and P. L. Toint, A primal-dual trust-region algorithm for non-convex nonlinear programming,, Mathematical Programming, 87 (2000), 215.
doi: 10.1007/s101070050112. |
[10] |
F. E. Curtis, A penalty-interior-point algorithm for nonlinear constrained optimization,, Mathematical Programming Computation, 4 (2012), 181.
doi: 10.1007/s12532-012-0041-4. |
[11] |
E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles,, Mathematical Programming, 91 (2002), 201.
doi: 10.1007/s101070100263. |
[12] |
C. Durazzi, On the Newton interior-point method for nonlinear programming problems,, Journal of Optimization Theory and Applications, 104 (2000), 73.
doi: 10.1023/A:1004624721836. |
[13] |
A. S. El-Bakry, R. A. Tapia, T. Tsuchiya and Y. Zhang, On the formulation and theory of the Newton interior-point method for nonlinear programming,, Journal of Optimization Theory and Applications, 89 (1996), 507.
doi: 10.1007/BF02275347. |
[14] |
R. Fletcher, Practical Methods of Optimization,, Second edition. A Wiley-Interscience Publication, (1987).
|
[15] |
P. E. Gill, W. Murray and M. H. Wright, Practical Optimization,, Academic Press, (1981).
|
[16] |
N. I. M. Gould, P. L. Toint and D. Orban, An Interior-point $l_1$-penalty Method for Nonlinear Optimization,, Groupe d'études et de recherche en analyse des décisions, (2010). Google Scholar |
[17] |
M. Guignard, Generalized Kuhn-Tucker conditions for mathematical programming problems in a Banach space,, SIAM Journal on Control, 7 (1969), 232.
doi: 10.1137/0307016. |
[18] |
X. X. Huang and X. Q. Yang, Convergence analysis of a class of nonlinear penalization methods for constrained optimization via first-order necessary optimality conditions,, Journal of Optimization Theory and Applications, 116 (2003), 311.
doi: 10.1023/A:1022503820909. |
[19] |
X. X. Huang and X. Q. Yang, A unified augmented Lagrangian approach to duality and exact penalization,, Mathematics of Operations Research, 28 (2003), 533.
doi: 10.1287/moor.28.3.533.16395. |
[20] |
X. W. Liu and J. Sun, A robust primal-dual interior-point algorithm for nonlinear programs,, SIAM Journal on Optimization, 14 (2004), 1163.
doi: 10.1137/S1052623402400641. |
[21] |
X. W. Liu and J. Sun, Global convergence analysis of line search interior-point methods for nonlinear programming without regularity assumptions,, Journal of Optimization Theory and Applications, 125 (2005), 609.
doi: 10.1007/s10957-005-2092-4. |
[22] |
Z. Q. Luo, J. S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints,, Cambridge University Press, (1996).
doi: 10.1017/CBO9780511983658. |
[23] |
K. W. Meng, S. J. Li and X. Q. Yang, A robust SQP method based on a smoothing lower order penalty functions,, Optimization, 58 (2009), 23.
doi: 10.1080/02331930701761193. |
[24] |
K. W. Meng and X. Q. Yang, First- and second-order necessary conditions via exact penalty functions,, Journal of Optimization Theory and Applications, 165 (2015), 720.
doi: 10.1007/s10957-014-0664-x. |
[25] |
K. W. Meng and X. Q. Yang, Optimality conditions via exact penalty functions,, SIAM Journal on Optimization, 20 (2010), 3208.
doi: 10.1137/090771016. |
[26] |
Z. Q. Meng, C. Y. Dang and X. Q. Yang, On the smoothing of the square-root exact penalty function for inequality constrained optimization,, Computational Optimization and Applications, 35 (2006), 375.
doi: 10.1007/s10589-006-8720-6. |
[27] |
M. Mongeau and A. Sartenaer, Automatic decrease of the penalty parameter in exact penalty function methods,, European Journal of Operational Research, 83 (1995), 686.
doi: 10.1016/0377-2217(93)E0339-Y. |
[28] |
A. S. Nemirovski and M. J. Todd, Interior-point methods for optimization,, Acta Numerica, 17 (2008), 191.
doi: 10.1017/S0962492906370018. |
[29] |
J. Nocedal and S. J. Wright, Numerical Optimization,, Springer Verlag, (2006).
|
[30] |
I. Pólik and T. Terlaky, Interior point methods for nonlinear optimization,, Nonlinear optimization, 1989 (2010), 215.
doi: 10.1007/978-3-642-11339-0_4. |
[31] |
R. T. Rockafellar and R. J. B. Wets, Variational Analysis,, Springer-Verlag, (1998).
doi: 10.1007/978-3-642-02431-3. |
[32] |
A. M. Rubinov, B. M. Glover and X. Q. Yang, Decreasing functions with applications to penalization,, SIAM Journal on Optimization, 10 (1999), 289.
doi: 10.1137/S1052623497326095. |
[33] |
D. F. Shanno and R. J. Vanderbei, Interior-point methods for nonconvex nonlinear programming: Orderings and higher-order methods,, Mathematical Programming, 87 (2000), 303.
doi: 10.1007/s101070050116. |
[34] |
G. Still and M. Streng, Optimality conditions in smooth nonlinear programming,, Journal of Optimization Theory and Applications, 90 (1996), 483.
doi: 10.1007/BF02189792. |
[35] |
R. J. Vanderbei and D. F. Shanno, An interior-point algorithm for nonconvex nonlinear programming,, Computational Optimization and Applications, 13 (1999), 231.
doi: 10.1023/A:1008677427361. |
[36] |
A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming,, Mathematical Programming, 106 (2006), 25.
doi: 10.1007/s10107-004-0559-y. |
[37] |
S. J. Wright, Primal-dual Interior-Point Methods,, SIAM, (1987).
doi: 10.1137/1.9781611971453. |
[38] |
Z. B. Xu, X. Y. Chang, F. M. Xu and H. Zhang, $L_{1/2}$ Regularization: A thresholding representation theory and a fast solver,, Neural Networks and Learning Systems, 23 (2012), 1013. Google Scholar |
[39] |
X. Q. Yang and Z. Q. Meng, Lagrange multipliers and calmness conditions of order $p$,, Mathematics of Operations Research, 32 (2007), 95.
doi: 10.1287/moor.1060.0217. |
[40] |
X. Q. Yang, Z. Q. Meng, X. X. Huang and G. T. Y. Pong, Smoothing nonlinear penalty functions for constrained optimization problems,, Numerical Functional Analysis and Optimization, 24 (2003), 351.
doi: 10.1081/NFA-120022928. |
[41] |
Y. Y. Ye, Interior Point Algorithms: Theory and Analysis,, Wiley-Interscience, (1997).
doi: 10.1002/9781118032701. |
show all references
References:
[1] |
H. Y. Benson, A. Sen and D. F. Shanno, Interior-point methods for nonconvex nonlinear programming: Convergence analysis and computational performance,, , (2009). Google Scholar |
[2] |
H. Y. Benson, D. F. Shanno and R. J. Vanderbei, Interior-point methods for nonconvex nonlinear programming: Jamming and numerical testing,, Mathematical Programming, 99 (2004), 35.
doi: 10.1007/s10107-003-0418-2. |
[3] |
R. H. Byrd, G. Lopez-Calva and J. Nocedal, A line search exact penalty method using steering rules,, Mathematical Programming, 133 (2012), 39.
doi: 10.1007/s10107-010-0408-0. |
[4] |
R. H. Byrd, J. Nocedal and R. A. Waltz, Steering exact penalty methods for nonlinear programming,, Optimization Methods and Software, 23 (2008), 197.
doi: 10.1080/10556780701394169. |
[5] |
R. Chartrand and V. Staneva, Restricted isometry properties and nonconvex compressive sensing,, Inverse Problems, 24 (2008).
doi: 10.1088/0266-5611/24/3/035020. |
[6] |
L. Chen and D. Goldfarb, Interior-point $l_2$-penalty methods for nonlinear programming with strong global convergence properties,, Mathematical Programming, 108 (2006), 1.
doi: 10.1007/s10107-005-0701-5. |
[7] |
L. Chen and D. Goldfarb, On the Fast Local Convergence of Interior-point $l_2$-penalty Methods for Nonlinear Programming,, Technical report, (2006). Google Scholar |
[8] |
X. J. Chen, Smoothing methods for nonsmooth, nonconvex minimization,, Mathematical Programming, 134 (2012), 71.
doi: 10.1007/s10107-012-0569-0. |
[9] |
A. R. Conn, N. I. M. Gould, D. Orban and P. L. Toint, A primal-dual trust-region algorithm for non-convex nonlinear programming,, Mathematical Programming, 87 (2000), 215.
doi: 10.1007/s101070050112. |
[10] |
F. E. Curtis, A penalty-interior-point algorithm for nonlinear constrained optimization,, Mathematical Programming Computation, 4 (2012), 181.
doi: 10.1007/s12532-012-0041-4. |
[11] |
E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles,, Mathematical Programming, 91 (2002), 201.
doi: 10.1007/s101070100263. |
[12] |
C. Durazzi, On the Newton interior-point method for nonlinear programming problems,, Journal of Optimization Theory and Applications, 104 (2000), 73.
doi: 10.1023/A:1004624721836. |
[13] |
A. S. El-Bakry, R. A. Tapia, T. Tsuchiya and Y. Zhang, On the formulation and theory of the Newton interior-point method for nonlinear programming,, Journal of Optimization Theory and Applications, 89 (1996), 507.
doi: 10.1007/BF02275347. |
[14] |
R. Fletcher, Practical Methods of Optimization,, Second edition. A Wiley-Interscience Publication, (1987).
|
[15] |
P. E. Gill, W. Murray and M. H. Wright, Practical Optimization,, Academic Press, (1981).
|
[16] |
N. I. M. Gould, P. L. Toint and D. Orban, An Interior-point $l_1$-penalty Method for Nonlinear Optimization,, Groupe d'études et de recherche en analyse des décisions, (2010). Google Scholar |
[17] |
M. Guignard, Generalized Kuhn-Tucker conditions for mathematical programming problems in a Banach space,, SIAM Journal on Control, 7 (1969), 232.
doi: 10.1137/0307016. |
[18] |
X. X. Huang and X. Q. Yang, Convergence analysis of a class of nonlinear penalization methods for constrained optimization via first-order necessary optimality conditions,, Journal of Optimization Theory and Applications, 116 (2003), 311.
doi: 10.1023/A:1022503820909. |
[19] |
X. X. Huang and X. Q. Yang, A unified augmented Lagrangian approach to duality and exact penalization,, Mathematics of Operations Research, 28 (2003), 533.
doi: 10.1287/moor.28.3.533.16395. |
[20] |
X. W. Liu and J. Sun, A robust primal-dual interior-point algorithm for nonlinear programs,, SIAM Journal on Optimization, 14 (2004), 1163.
doi: 10.1137/S1052623402400641. |
[21] |
X. W. Liu and J. Sun, Global convergence analysis of line search interior-point methods for nonlinear programming without regularity assumptions,, Journal of Optimization Theory and Applications, 125 (2005), 609.
doi: 10.1007/s10957-005-2092-4. |
[22] |
Z. Q. Luo, J. S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints,, Cambridge University Press, (1996).
doi: 10.1017/CBO9780511983658. |
[23] |
K. W. Meng, S. J. Li and X. Q. Yang, A robust SQP method based on a smoothing lower order penalty functions,, Optimization, 58 (2009), 23.
doi: 10.1080/02331930701761193. |
[24] |
K. W. Meng and X. Q. Yang, First- and second-order necessary conditions via exact penalty functions,, Journal of Optimization Theory and Applications, 165 (2015), 720.
doi: 10.1007/s10957-014-0664-x. |
[25] |
K. W. Meng and X. Q. Yang, Optimality conditions via exact penalty functions,, SIAM Journal on Optimization, 20 (2010), 3208.
doi: 10.1137/090771016. |
[26] |
Z. Q. Meng, C. Y. Dang and X. Q. Yang, On the smoothing of the square-root exact penalty function for inequality constrained optimization,, Computational Optimization and Applications, 35 (2006), 375.
doi: 10.1007/s10589-006-8720-6. |
[27] |
M. Mongeau and A. Sartenaer, Automatic decrease of the penalty parameter in exact penalty function methods,, European Journal of Operational Research, 83 (1995), 686.
doi: 10.1016/0377-2217(93)E0339-Y. |
[28] |
A. S. Nemirovski and M. J. Todd, Interior-point methods for optimization,, Acta Numerica, 17 (2008), 191.
doi: 10.1017/S0962492906370018. |
[29] |
J. Nocedal and S. J. Wright, Numerical Optimization,, Springer Verlag, (2006).
|
[30] |
I. Pólik and T. Terlaky, Interior point methods for nonlinear optimization,, Nonlinear optimization, 1989 (2010), 215.
doi: 10.1007/978-3-642-11339-0_4. |
[31] |
R. T. Rockafellar and R. J. B. Wets, Variational Analysis,, Springer-Verlag, (1998).
doi: 10.1007/978-3-642-02431-3. |
[32] |
A. M. Rubinov, B. M. Glover and X. Q. Yang, Decreasing functions with applications to penalization,, SIAM Journal on Optimization, 10 (1999), 289.
doi: 10.1137/S1052623497326095. |
[33] |
D. F. Shanno and R. J. Vanderbei, Interior-point methods for nonconvex nonlinear programming: Orderings and higher-order methods,, Mathematical Programming, 87 (2000), 303.
doi: 10.1007/s101070050116. |
[34] |
G. Still and M. Streng, Optimality conditions in smooth nonlinear programming,, Journal of Optimization Theory and Applications, 90 (1996), 483.
doi: 10.1007/BF02189792. |
[35] |
R. J. Vanderbei and D. F. Shanno, An interior-point algorithm for nonconvex nonlinear programming,, Computational Optimization and Applications, 13 (1999), 231.
doi: 10.1023/A:1008677427361. |
[36] |
A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming,, Mathematical Programming, 106 (2006), 25.
doi: 10.1007/s10107-004-0559-y. |
[37] |
S. J. Wright, Primal-dual Interior-Point Methods,, SIAM, (1987).
doi: 10.1137/1.9781611971453. |
[38] |
Z. B. Xu, X. Y. Chang, F. M. Xu and H. Zhang, $L_{1/2}$ Regularization: A thresholding representation theory and a fast solver,, Neural Networks and Learning Systems, 23 (2012), 1013. Google Scholar |
[39] |
X. Q. Yang and Z. Q. Meng, Lagrange multipliers and calmness conditions of order $p$,, Mathematics of Operations Research, 32 (2007), 95.
doi: 10.1287/moor.1060.0217. |
[40] |
X. Q. Yang, Z. Q. Meng, X. X. Huang and G. T. Y. Pong, Smoothing nonlinear penalty functions for constrained optimization problems,, Numerical Functional Analysis and Optimization, 24 (2003), 351.
doi: 10.1081/NFA-120022928. |
[41] |
Y. Y. Ye, Interior Point Algorithms: Theory and Analysis,, Wiley-Interscience, (1997).
doi: 10.1002/9781118032701. |
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